For a population with μ = 60 and σ = 8, the z-score corresponding to X = 72 would be ___.

2. A normal distribution has a mean of μ = 100 and σ = 20. If one score is randomly selected from this distribution, what is the probability that the score will have a value between X = 90 and X = 120.

3. A distribution is normal and has μ = 60 and σ = 10. What is the 40th percentile? (Hint: Use table to find z score and math to find raw score)

4. A random sample of n = 4 scores is obtained from a population with a mean of has μ = 80 and σ = 10. If the sample mean is M = 90, what is the z-score for the sample mean?

For 5-6: Assume a normal distribution with μ = 70 and σ = 12. Use the Z-table in the back of this exam to answer the following questions. (Show distributions and work)
5. What percentage of scores are above 80? (4 pts)

6. What percentage of scores are between 56 and 66? (5pts)

7. Assume a normal distribution of test scores with μ = 50 and σ = 10. Use the Z-table in the back of this exam to answer the following question: What would be the percentile ranking for someone with a score of 56? (Show distribution and work)

8. Suppose you selected samples of 9 persons (so n=9) at random who take a test with an average of μ = 100 and σ = 15. What proportion of all possible samples of size n=9 would be expected to score less than M = 106? (Hint: Use z-score formula that takes into account n.)(4 pts)

For 9-11: Assume that a group of students show up for the first day of class and are given a test. Assume that students must completely guess on every question. The test includes 50 multiple choice questions and each question has 5 possible responses. Each question is scored as correct or incorrect, and hence, follows a binomial distribution. Use the z approximation to the binomial to solve.

9. On average, how many of the 40 test questions do you expect to students to get right simply guessing?

10. What would the standard deviation for this be?

11. What would be the probability of getting a score of 13 or greater? (Hint: Use real limits to capture all the area for a score of 10 for this problem only.)

12-13. A researcher draws all possible unique samples of size n = 144 from a college population and collects data on the ages of the subjects where μ = 50 and σ = 20. She constructs a distribution of the mean ages computed for each sample.

12. What is the expected value of the mean of these sample means? (Write answer using the notation for the expected value is = to __.)(2pts)

13. What is the value of the standard error of the mean? (2pts)

Z-TEST

A researcher is interested in evaluating the effectiveness of using Ritalin to improve test scores among children with ADHD. Assume that we know that, on average, children with untreated ADHD score μ = 65 on a standardized math test and the standard deviation is
σ = 18. The researcher draws a random sample of size n = 64 of children with ADHD and treats them for a year with Ritalin. At the end of the year, the average improvement among the sample of those receiving Ritalin is found to be M= 70. Does Ritalin significantly increase test performance? Conduct a two-tailed Z-test with alpha set at .05 to test the hypothesis that the treatment is effective.

16. Step 2: Set the criteria for making a decision. Note alpha (for one or two tailed test). Make a diagram of the regions of acceptance and rejection associated with the null hypothesis, labeling the horizontal axis using Z-score values. (6pts)

17. Step 3: Calculate the Z-statistic; use the following information that the M = 70, where μ = 65, σ = 18, and n = 64.

18-19. Step 4: Make your decision. 18. Given your answer in step 3, would you accept or reject the null hypothesis? 19. Briefly state your conclusion (in your own words) regarding the effectiveness of the new treatment for depression?

20. Construct a 95% confidence interval to estimate the location of the true population mean for those who received the Ritalin treatment.